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A generalization of Grünbaum's inequality in RCD$(0,N)$-spaces

Victor-Emmanuel Brunel, Shin-ichi Ohta, Jordan Serres

TL;DR

The paper advances Grünbaum's inequality into the synthetic setting of RCD(0,N) spaces and weighted manifolds with nonnegative Ricci curvature, showing that a convex set Ω with barycenter x0 forces a uniform lower bound on the mass of Ω intersected with horoballs defined by Busemann functions along any straight line. The main methodology blends Gigli's splitting theorem with Cavalletti–Mondino localization to reduce problems to a one-dimensional analysis, from which sharp bounds and rigidity statements follow. The results cover both β-concavity/β-convexity regimes and exponential bounds in the log-density case, and they include stability estimates describing how near-equality configurations approximate model cones. These contributions extend classical Grünbaum-type inequalities to curved and non-smooth spaces, highlighting the role of 1D reduction, cone-structure rigidity, and quantitative stability in synthetic geometry.

Abstract

We generalize Grünbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to $\mathrm{RCD}(0,N)$-spaces with $N \in (1,\infty)$ as well as weighted Riemannian manifolds of $\mathrm{Ric}_N \ge 0$ for $N \in (-\infty,-1) \cup \{\infty\}$. Our formulation makes use of the isometric splitting theorem; given a convex set $Ω$ and the Busemann function associated with any straight line, the volume of the intersection of $Ω$ and any sublevel set of the Busemann function that contains a barycenter of $Ω$ is bounded from below in terms of $N$. We also extend this inequality beyond uniform distributions on convex sets. Moreover, we establish some rigidity results by using the localization method, and the stability problem is also studied.

A generalization of Grünbaum's inequality in RCD$(0,N)$-spaces

TL;DR

The paper advances Grünbaum's inequality into the synthetic setting of RCD(0,N) spaces and weighted manifolds with nonnegative Ricci curvature, showing that a convex set Ω with barycenter x0 forces a uniform lower bound on the mass of Ω intersected with horoballs defined by Busemann functions along any straight line. The main methodology blends Gigli's splitting theorem with Cavalletti–Mondino localization to reduce problems to a one-dimensional analysis, from which sharp bounds and rigidity statements follow. The results cover both β-concavity/β-convexity regimes and exponential bounds in the log-density case, and they include stability estimates describing how near-equality configurations approximate model cones. These contributions extend classical Grünbaum-type inequalities to curved and non-smooth spaces, highlighting the role of 1D reduction, cone-structure rigidity, and quantitative stability in synthetic geometry.

Abstract

We generalize Grünbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to -spaces with as well as weighted Riemannian manifolds of for . Our formulation makes use of the isometric splitting theorem; given a convex set and the Busemann function associated with any straight line, the volume of the intersection of and any sublevel set of the Busemann function that contains a barycenter of is bounded from below in terms of . We also extend this inequality beyond uniform distributions on convex sets. Moreover, we establish some rigidity results by using the localization method, and the stability problem is also studied.
Paper Structure (26 sections, 23 theorems, 125 equations)

This paper contains 26 sections, 23 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Notations and definitions
  4. Metric geometry.
  5. Curvature-dimension conditions.
  6. Splitting theorems.
  7. Main results
  8. Preliminaries
  9. Properties of $\mathrm{RCD}(0,N)$-spaces
  10. Euclidean case
  11. One-dimensional analysis
  12. One-dimensional $\mathrm{CD}(0,N)$-spaces
  13. Case of $N \in (1,\infty)$
  14. Case of $N \in (-\infty,-1) \cup \{\infty\}$
  15. Grünbaum’s inequality for $N \in (1,\infty)$
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $(X,\mathrm{d}_X,\mathfrak{m})$ be an $\mathrm{RCD}(0,N)$-space with $N \in (1,\infty)$, $\mu=\rho\mathfrak{m} \in \mathcal{P}^1(X)$ with a measurable function $\rho\colon X \longrightarrow [0,\infty)$, and $x_0 \in X$ be any barycenter of $\mu$. Suppose that there is a straight line $\gamma\col

Theorems & Definitions (48)

  • Theorem 1.1: Main theorem; $N>1$
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5: Main theorem; $N=\infty$, $N<-1$
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 38 more